Opuscula Math. 26, no. 2 (2006), 243-256

 
Opuscula Mathematica

Application of Green's operator to quadratic variational problems

Nikolay V. Azbelev
Vadim Z. Tsalyuk

Abstract. We use Green's function of a suitable boundary value problem to convert the variational problem with quadratic functional and linear constraints to the equivalent unconstrained extremal problem in some subspace of the space \(L_2\) of quadratically summable functions. We get the necessary and sufficient criterion for unique solvability of the variational problem in terms of the spectrum of some integral Hilbert-Schmidt operator in \(L_2\) with symmetric kernel. The numerical technique is proposed to estimate this criterion. The results are demonstrated on examples: 1) a variational problem with deviating argument, and 2) the problem of the critical force for the vertical pillar with additional support point (the qualities of the pillar may vary discontinuously along the pillar's axis).

Keywords: quadratic variational problem, Sobolev space, boundary value problem, Hilbert space, Green's operator, Fredholm integral operator, spectrum.

Mathematics Subject Classification: 49N10, 49K27, 34B27, 47G10.

Full text (pdf)

Opuscula Mathematica - cover

Cite this article as:
Nikolay V. Azbelev, Vadim Z. Tsalyuk, Application of Green's operator to quadratic variational problems, Opuscula Math. 26, no. 2 (2006), 243-256

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.